Inequalities relating to L_p-version of Petty's conjectured projection inequality

摘要

Petty's conjectured projection inequality is a famous open problem in the theory of convex bodies.In this paper,it is shown that an inequality relating to L_p-version of the Petty's conjectured projection inequality is developed by using the notions of the L_p-mixed volume and the L_p-dual mixed volume,the relation of the L_p-projection body and the geometric bodyΓ_(-p)K,the Bourgain-Milman inequality and the L_p-Busemann- Petty inequality.In addition,for each origin-symmetric convex body,by applying the Jeusen inequality and the monotonicity of the geometric bodyΓ_(-p)K,the reverses of L_p-version of the Perry's conjectured projection inequality and the L_p-Petty projection inequality are given,respectively.